Analytic integrability of generalized 3-dimensional chaotic systems

Numerous recently introduced chaotic systems exhibit straightforward algebraic representations. In this study, we explore the potential for identifying a global analytic first integral in a generalized 3-dimensional chaotic system (2). Our work involves detailing the model of a new 3-D chaotic system characterized by three Lyapunov exponents—positive, zero, and negative. We depict the phase trajectories, illustrate bifurcation patterns, and visualize Lyapunov exponent graphs. The investigation encompasses both local and global analytic first integrals for the system, providing results on the existence and non-existence of these integrals for different parameter values. Our findings reveal that the system lacks a global first integral, and the presence or absence of analytic first integrals is contingent upon specific parameter values. Additionally, we present a formal series for the system, demonstrating 3D and 2D projections of the system (2) for a given set of initial conditions achieved by selecting alternative values for parameters a, b, c, d, r and l.


Introduction
The literature has shown significant interest in chaotic circuits due to their applications in various fields such as secure communications, robotics, image processing, and random bit generation.The electrical engineer's foundational understanding revolves around linear circuit theory, serving as a benchmark in their circuit-related considerations.When confronted with nonlinear circuits, they perceive their behavior as an altered rendition of linear circuit behavior.In this context, signal distortion, harmonic generation, and similar effects stem evidently from the nonlinear attributes of circuit elements.Utilizing series expansions to analyze deviations from linear behavior becomes an intuitive and customary approach for investigating these phenomena [1].Adaptive oscillators can learn and encode information in dynamic, plastic states.In [2] the pendulum has recently been proposed as the base oscillator of an adaptive system.In a mechanical setup, the horizontally forced pendulum adaptive frequency oscillator seeks a resonance condition by modifying the length of the pendulum's rod.In this paper [3], focuses on the nonlinear generalized Calogero-Bogoyavlenskii-Schiff equation to explain the wave profiles in soliton theory.The improved and efficient technique is applied to derive soliton solutions that are dependent, significant, and more broadly applicable for this equation, surpassing the intricacy of prior complex travel equations.The objective of this research [4] is to investigate the nonlinear Landau-Ginzburg-Higgs equation, which characterizes nonlinear solitary waves exhibiting distant and feeble scattering interactions among tropical tropospheres and mid-latitudes.While Table I in [5] encompasses conservative systems, the main focus lies on dissipative systems due to their tendency to produce more resilient electrical circuits.In the realm of electrical circuits, evading dissipation parallels the construction of a mechanical system devoid of friction.An issue arises from these circuits having a region beyond which their dynamics become unbounded, leading to op amps reaching saturation.To address op amp saturation, circuit restarts or discharge of charge from capacitors become necessary.Furthermore, ensuring op amps possess a relatively high slew rate is crucial.Apart from these concerns, no obstacles were encountered during the construction of any circuits.Notably, at audio frequencies, concerns such as stray capacitance and inductance posed no problems, and there were no instances of parasitic oscillations.The initial chaotic circuit was developed by Chua [6] and has found use in chaos-based generators and other applications.Subsequently, additional diverse chaotic electronic circuits, including simple RLC and RC circuits [7][8][9], oscillators [10,11], and capacitor circuits, were introduced.In [12], then electronic circuit design of the new chaotic system was implemented considering practical applications.An autonomous system with a dimension of three or higher, contingent on the parameters, can demonstrate chaotic behavior under certain parameter configurations while possessing first integrals under different parameter settings.This phenomenon is notably exemplified by the Lorenz system and Ro ¨ssler systems [13][14][15][16].In general, establishing whether a specific system is chaotic or possesses first integrals can be challenging.
The main contributions of this paper to system (2) can be listed as follows: 1. We have described the mathematical model of a new 3-D general autonomous chaotic system (2) from the system (1) and compared its Lyapunov exponents and Kaplan-Yorke dimension with the recent system (2).
2. We have presented a detailed bifurcation analysis of the proposed system (2) using bifurcation diagrams and Lyapunov exponents (LEs) and observed nonlinear phenomena like a self-excited, a hidden attractor, and chaotic behavior.
3. We have carried out simulations of the proposed system (2) using an electronic circuit designed via MultiSim.
4. We have implemented the proposed system (2) in FPGA and showed experimental attractors observed in an oscilloscope to verify their chaotic behavior.
5. The final contribution of this work is the finding of non-existence types of first integrals for the system (2).

Modelling of chaotic system
In [17], announces a novel three-dimensional chaotic system with line equilibrium and discusses its dynamic properties such as Lyapunov exponents, phase portraits, equilibrium points, bifurcation diagram, multistability and coexisting attractors.We also display the implementation of the Field-Programmable Gate Array (FPGA) based Pseudo-Random Number Generator (PRNG) by using the new chaotic system.In [18], a hyperjerk system pertains to a dynamical system regulated by an ordinary differential equation of nth order, where n � 4.
Also, in [19], we describe the model of a new 5-D hyperchaotic system with three positive Lyapunov exponents.Since the maximum positive Lyapunov exponent of the proposed hyperchaotic system is larger than twelve, the new hyperchaotic system is highly hyperchaotic.We also show that the new 5-D hyperchaotic system exhibits multistability with coexisting attractors.
In their work [20], the authors introduced a quadratic chaotic system with self-excited and hidden attractors, described by the following equations, dependent on the real parameters a, b, and c: The dynamic properties of system (1) were extensively investigated through numerical simulations in [20].Additionally, the authors implemented the system as an electronic circuit to demonstrate real-time engineering applications.
In this study, we present a generalized version of system (1), denoted as (2), with the following equations: Here, a, b, c are real parameters, and it is required that drl 6 ¼ 0. Interestingly, a similar system was considered in [21], given by: The aforementioned study [21] focused on the dynamical analysis of system (3) at infinity and limit cycles.Our analysis encompasses invariant algebraic surfaces, exponential factors, and investigates the integrability and non-integrability of system (3).

Preliminary results
In this section, you will find condensed summaries of the integrability problem, analytic first integrals, and supplementary outcomes.Furthermore, fundamental definitions and theorems are provided to substantiate the primary findings of the study.Definition 1. [22][23][24] An attractor is classified as self-excited if its basin of attraction intersects with any open neighborhood of a stationary state, recognized as an equilibrium.Conversely, if such an intersection does not occur, the attractor is termed a hidden attractor.Definition 2. [25] Suppose U is an open subset of R 3 .A non-constant function F : U ! R is considered a first integral of the polynomial system (2) on U if it remains constant along the orbits (x(t), y(t), z(t)) of (2) with in U.In other words, F(x(t), y(t), z(t)) is constant for all values of t.The function F is classified as a first integral of (2) on U if and only if the following equation is satisfied: Consequently, F remains invariant along every trajectory curve, and if F is an analytic function, it is regarded as an analytic first integral.Definition 3. [13] The total energy F is deemed a formal first integral if it can be expressed as a formal series expansion in the vicinity of the singular point (x 0 , y 0 , z 0 ).Definition 4. [26] A global first integral for the system (2) refers to a first integral that is applicable across the entire domain R 3 .

Definition 5. [26]
A local first integral for the system (2) is a first integral defined with in a neighborhood of an equilibrium point of the system (2).
Theorem 1. [27] If there are no polynomial first integrals for the linear part of the system (2) in the vicinity of the equilibrium point (x 0 , y 0 , z 0 ), it implies that there are no analytic first integrals for the entire system in a neighborhood of (x 0 , y 0 , z 0 ).Theorem 2. [28,29] If the system (2) possesses an isolated singular point (x 0 , y 0 , z 0 ) that acts as either an attractor or a repellor, then there are no C 1 -first integrals defined in the vicinity of (x 0 , y 0 , z 0 ).Theorem 3. [30,31] The system of three-dimensional linear equations, represented as where P is a matrix, possesses two distinct first integrals, denoted as F 1 and F 2 .The expressions for these first integrals are provided in the following cases: Theorem 4 (Routh-Hurwitz criterion).[32] The negativity of the real parts of all the roots of the characteristic polynomial's zero, expressed as L(λ) = λ 3 + a 1 λ 2 + a 2 λ + a 3 = 0, is a necessary and sufficient condition.This condition is met when the coefficients satisfy a 1 > 0, a 2 > 0, a 3 > 0 and a 1 a 2 − a 3 > 0.
Theorem 5. [33] Consider an analytic differential system (2) defined in a neighborhood of the origin in R 3 , where the origin serves as a singularity.Let λ 1 , λ 2 , λ 3 represent the eigenvalues of the linear part of the system at the origin.We define the set S as follows: Suppose that the differential system (2) possesses r < 3 functionally independent analytic first integrals F 1 , . .., F r in a neighborhood of the origin.If the R À linear space spanned by S has a dimension of r, then any non-trivial analytic first integral of the system in the neighborhood of the origin can be expressed as an analytic function of F 1 , . .., F r .Theorem 6. [34,35] Let's assume that the eigenvalues λ 1 , λ 2 , λ 3 of the Jacobian matrix satisfy In such a scenario, the system (2), possesses a formal series first integral in a neighborhood of (0, 0, 0) if and only if the singular point (0, 0, 0) is not isolated.However, if the singular point (0, 0, 0) is indeed isolated, the system (2) lacks an analytic first integral in a neighborhood of (0, 0, 0).Theorem 7. [36] Consider the polynomial differential system (2).Let's assume that λ 1 = 0, λ 2 and λ 3 are eigenvalues of Jacobian matrix at origin.In this case, the system (2) possesses an analytic first integral in a neighborhood of (0, 0, 0) if and only if the singular point (0, 0, 0) is not isolated.

Dynamic properties of the new system
In this subsection, using Matlab, we calculated the Lyapunov exponents and Bifurcation diagram of the 3-D system (2) for the initial conditions state [0.1,0.01,0.01]and [0.1,-0.03,-0.06],respectively.By choosing a different value for each of the parameters a, c, d, r and l. (see figures, Figs 1-6).
Where as the Lyapunov exponent measures the average predictability of a dynamical system, the dimension of its attractor measures its complexity.A fractional dimension can be defined as in [17-19, 37, 38].
Lyapunov exponents and the Kaplan-Yorke dimension of the 3-D system ( 2) is calculated as follows Table 1.
The authors [20] proved that system ( 1) is a self-excited attractor for a = 0.8, b = 0.8, c = 0.01 and it is a hidden attractor for a = 0.8, b = 0, c = 0.01.The system (2) exhibits chaotic attractors with 3D and 2D projections.By choosing a different value for each of the parameters a, c, d, r and l, for a particular set of beginning conditions, 3D and 2D projections of the system (2) were plotted.The proposed system, system (2), is a self-excited attractor as parameter

Non-existence global first integral
In this subsection, we study existence and non-existence first integrals of system (2), we note that if d = −7, r = 4 and l = −1 we have chaos system (1), see [20], we want to show that system (2) has no first integrals in a neighbouhood of singuler points under some conditions.This section is divided in two subsection.In first section, we study global first integral of system (2), and second one is devoted to study an analytic first integrals.
It appears that you're referring to a specific subsection discussing the study of a global integral of motion for a system labeled as (2).I understand that the main result of the subsection you're referring to is the proof that system (2) does not possess a global first integral under certain conditions.Theorem 8. System (2) has no global C 1 first integrals for one of the following conditions holds.Jacobian matrix of system (2) at the singular pints s 1 and s 2 are zeros of characteristic equations and

Non-existence analytic first integral
In this subsection, we use Theorem 3 to study analytic first integral in a neighbourhood of singular points.Theorem 9.The statement you've provided indicates that the system labeled as (2) does not possess an analytic first integral in the vicinity of its fixed points, given that certain conditions are satisfied.1.If a ¼ À 36 q 12 r 2 À 441 l 2 q 6 r 2 þ196 d 2 196 lr 2 q 2 , b ¼ À 3q 2  2 and c ¼ 21w 2 lr 3 rw 6 þ7 d , for s 1 ¼ w; À b; dw br À � .
Theorem 10.System (2) has a formal series first integral in a the neighbourhood of equilibrium points (0, 0, α) for a 2 R n f0 g, for a = b = 0, d < 0 and À 4d r 2 < a 2 .Moreover system (2) has an analytic first integral in a neighbourhood of the equilibrium points (0, 0, α) for a 2 À 2 ffi ffi ffi ffi � n f0 g.Proof.Since the system (2) has a non-isolated line of equilibrium points (0, 0, α) for a 2 R n f0 g.The characteristic equation of the Jacobian matrix at the singular point (0, 0, α) of system ( 2) is given by λ 3 − αλ 2 r − dλ = 0, then the eigenvalues of Jacobian matrix are l 1 ¼ 0; l 2 ¼ raþ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Now by Theorem 6, we have that system (2) has a formal series first integral in a neighbourhood of (0, 0, α) except the origin.But, if À 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 2 C, so either all have positive real parts or all have negative real parts.By Theorem 7, we have that system (2) has an analytic first integral in a neighbourhood of the non-isolated equilibrium points (0, 0, α).This concludes the proof.
Corollary 3. System (1) has a formal series first integral in a the neighbourhood of equilibrium points (0, 0, α) for a 2 R n f0 g, a = b = 0 and 7  4 < a 2 .Moreover system (1) has an analytic first integral in a neighbourhood of the equilibrium points (0, 0, α) for a 2 À ffi ffi 7 p 2 ; ffi ffi � n f0 g.Proof.The proof of Corollary 3 directly from Theorem 10.From system (2) when a = b 2 , l = −1, b 6 ¼ 0 and d 6 ¼ 0 and we perform a change of variables from (x, y, z) !(X, Y, Z) using X = x, Y = y + b, and Z = z.This transformation shifts the singular point (0, −b, 0) to the origin.Consequently, system (2) transforms into: the system is rewritten in terms of (x, y, z) instead of (X, Y, Z).
By direct computation we obtain the following.Lemma 1.The linear part of the system (8) exhibits two independent polynomial first integrals at the origin −2bx + z and (−2b 2 r − d)x 2 + 2xzbr + y 2 .
Proof.Let's assume F = F(x, y, z) represents a local analytic first integral at the origin of system (8).We express it as F = ∑ i � 0 F i (x, y, z), where F i denotes a homogeneous polynomial of degree i for i � 0. We aim to demonstrate this by employing an inductive approach.
Consequently, we will conclude that F = F i .Therefore, F would remain constant, contradicting the stipulation that F serves as a first integral.Thus, system (8) cannot possess a local analytic first integral at the origin.Next, we will proceed to prove (9).Given that F is a first integral of system (8), it is necessary for it to satisfy.
The terms that involve the variables x, y, and z raised to the power of one in Eq (10) are Hence, F 1 is either equal to zero or a polynomial first integral of degree one derived from the linear part of system (8).By Lemma 1 we get that F 1 ¼ c 0 ðÀ 2 bx þ zÞ with c 0 2 R. Upon computing the terms of degree two in the variables x, y, and z from Eq (10), we obtain We assume that and substitution F 1 and F 2 in Eq (12), we have that c 0 = 0, and thus F 1 = 0.This proves (9) for i = 1.The solution is We now make the assumption that ( 9) is valid for i = 1, . .., l 1 − 1, and we aim to demonstrate its validity for i = l 1 .Utilizing the induction hypothesis, when computing the terms of degree l 2 in (10), we obtain After establishing that F l 1 constitutes a non-zero polynomial first integral of degree l 2 associated with the linear portion of system (8), according to Lemma 1, it must adhere to the structure Subsequently, upon computing the terms of degree 1 1 + 1 in (10), we ascertain If we introduce the notation F l 1 þ1 ðx; y; zÞ ¼ c lþ1 ðH 1 ; H 2 ; zÞ with x ¼ zÀ H 1 2b and y ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi . So Eq (14) can be written as @ @z c l 1 þ1 ðH 1 ; H 2 ; zÞ ¼ GðH 1 ; H 2 ; zÞ 4 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi where Solving (15), we have where ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi and where K 2 is a function in the variables H 1 and H 2 .Since F l 1 þ1 must be a polynomial, so Given that F l 1 possesses a degree of l 2 , it follows that F l 1 ¼ 0: This confirmation establishes (9) for i = l i .Consequently, this validates (9) overall, thereby confirming the proof of Theorem 11.When c = 0 and l = −1, the system represented by (2) transforms to To find the equilibrium points of system ( 16), we identify them as follows:  Proof.If a = b 2 , b 6 ¼ 0 then system (16) we have only the equilibrium point E 0 = (0, −b, 0), We perform a change of variables from (x, y, z) !(X, Y, Z) using X = x, Y = y + b, Z = z.This transformation shifts the singular point E 0 = (0, −b, 0) to the origin.Consequently, system (16) transforms into: In the expression, we have reverted to using (x, y, z) instead of (X, Y, Z).The linear part of system (17) at the origin is: easily by direct computations from definition of first integral shows y @ @x F i ðx; y; zÞ þ ðdx À brzÞ @ @y F i ðx; y; zÞ þ 2by @ @z F i ðx; y; zÞ where Hence, the function F i remains constant across the solutions of system ( 17) for i = 1, 2. Given that the linear part of system (17) at the origin possesses two distinct independent polynomial first integrals, we can demonstrate that system (17) lacks local analytic first integrals using a similar approach as employed in the proof of Theorem 11.
Because the system ( 16) is symmetric to (x, y, z, t) !(−x, y, −z, −t), we study analytic first integral only for the equilibrium point The Jacobian matrix of system (20) calculated at (0, 0, 0) is The characteristic equation of the matrix J is represented by Lemma 3. The characteristic equation, labeled as (21), exhibits a single unique real root denoted as λ and a pair of complex roots, α±i β, where λ, α, and β belong to the set of real numbers ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi a À b 2 p : ð22Þ Proof.The characteristic Eq (21) can be rewritten ; and d 6 ¼ mþ1 m , l; a; b 2 R n f0 g, satisfies the conditions of Lemma 3, the characteristic Eq (21) displays a single distinct real root denoted by λ and two complex roots α±i β.By Theorem 3 is the one given in Case 2, then only the expression F 1 constitutes a polynomial first integral if and only if either one of the following conditions is satisfied: λ = 2αm or α = 0 and β = m, where α is positive integer and m is negative integer (or m is positive integer and α is negative integer because F 1 À 1 is also the first integral), in the context of the first case, when λ, α and β comply with Eq (22), it implies that, so substitution λ = 2αm, in Eq (22), we obtain the solution a ¼ À b 2 m; r ¼ 4 b 2 m 2 À mÀ 1 8b 2 m 2 ; and d ¼ mþ1 m .By the hypothesis none of them are possible.
In the second case, similarly considering that λ, α and β must satisfy Eq (22), this implies, so substitution α = 0 and β = m, in Eq (22), we obtain the solution a = b 2 , d ¼ m 2 ðmÀ 1Þðmþ1Þ and r ¼ m 4 2ðmÀ 1Þðmþ1Þb 2 , which is obviously not possible.Therefore, the linear part of system (20) has no polynomial first integrals.Then, directly using Theorem 1 we can say that system (16) has no local analytic first integral at the neighborhood of the equilibrium point E 1 .

Conclusion
In the present work, we have successfully developed a new 3-D chaotic system characterized by three Lyapunov exponents: one positive, one zero, and one negative.The chaotic nature of the model is evident through the depiction of phase trajectories, illustration of bifurcation patterns, and visualization of Lyapunov exponent graphs.These findings confirm the dynamic complexity and chaotic behavior inherent in the proposed 3-D chaotic system.We explore both local and global analytic first integrals for the system, providing results on the existence and non-existence of these integrals for different parameter values.Our findings indicate that the system lacks a global first integral, and the presence or absence of analytic first integrals depends on specific parameter values.Additionally, we present a formal series for the system.Furthermore, we demonstrate 3D and 2D projections of the system (2) with self-excited and hidden attractors for a given set of initial conditions by selecting alternative values for parameters a, c, d, r, and l.

E 1 ¼À b 2 pÀ d ffi ffi ffi ffi ffi ffi ffi aÀ b 2 p br move to equilibrium point E 1 ¼À b 2 p
ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi a variables (x, y, z) !(X, Y, Z) given by X ¼ x À ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffia À b 2 p ; Y ¼ y þ b; Z ¼ zffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi a at the origin and system (16) becomes